Download 8.3

Chapter 8: Informal Geometry
Section 8.3: Simple Closed Curves - Answers
Simple Closed Curves
•
Plane curve: a set of points that can be drawn without lifting the pencil
•
Simple: without lifting the pencil , do not retrace any of its points
•
Closed curve: drawn by starting and stopping at the same point
•
Jordan Curve Theorem: A simple closed curve c separates a plane into three disjoint sets of points
called the interior, the exterior, and the curve itself .
•
Region: Union of simple closed curve and its interior .
•
Convex Region: a region is convex if, for any two points in the region, the line segment joining them
lies completely in the region.
•
Nonconvex (concave) region: a region is concave if, for any two points in the region, the line segment
joining them does not lie completely in the region.
•
Convex curves: Curves bounding convex regions are called convex curves.
Circles
•
Circle: the set of all points in a plane at a given distance from a fixed point in the plane is a circle.
•
Center: the fixed point in the center of a circle
•
Radius: a line segment connecting the center to a point on the circle
•
Diameter: a line segment connecting two points on the circle, containing the center
•
Chord: a line segment connecting two points on the circle, not containing the diameter
•
Disk: the area of a circle plus the points of the circle
Polygons
•
Polygon: a polygon is a simple closed curve that is the union of three or more line segments, such that
all the points are coplanar and distinct, no three consecutively named points are collinear.
•
Sides: the line segments that form the polygon are called sides
•
Vertices: the points defining the sides are called vertices
•
Adjacent vertices: endpoints on the same side (line segment)
•
Diagonals: line segments joining non-adjacent vertices
•
Adjacent sides: sides with a common vertex
•
Interior angles: we refer the angles of a convex polygon as the interior angles
•
Regular polygon: a simple convex polygon with all sides of equal length and all angles of equal
measure is called a regular polygon. Such a polygon is said to be equilateral and equiangular.
•
Exterior angles: formed by extending one side of a polygon and measuring the angle formed; the
interior angle and the exterior angle are supplementary to each other, their measures sum to 180
degrees
Regular Polygons or n-gons (n sides):
Name
Number of Sides
1. Triangle
3
Number of angles
3
2. Quadrilateral
4
4
3. Pentagon
5
5
4. Hexagon
6
6
5. Heptagon or 7-gon
7
7
6. Octagon
8
8
Triangles
•
Equilateral: all three sides are of equal length and all 3 angles are of equal measure
•
Isosceles: at least two sides are congruent (have equal length) ; the angles opposite these sides are also
congruent
•
Scalene: no two sides are congruent (no two sides have equal length)
Quadrilaterals
Quadrilateral Definitions:
1. A parallelogram is a quadrilateral with both pairs of opposite sides parallel
2. A trapezoid is a quadrilateral with exactly one pair of opposite sides parallel
3. A rectangle is a parallelogram with a right angle. One right angle implies that all four angles are right
angles.
4. A square is a rectangle with all sides of equal length.
5. A rhombus is a parallelogram with all sides of equal length.
6. A kite is a quadrilateral with two distinct pairs of adjacent sides of equal length
Angles of Polygons
•
Interior angles – any polygon: may all be of different measures
•
Interior angles – regular polygon: are ALWAYS equal measures
•
Central angle: formed by connecting the center of a regular polygon with a pair of adjacent vertices
Angle Measures of Regular n-gons:
Number of sides Angle Sum
1.
3
1 ∙ 180°
Measure of Vertex Angle
∙
°
60°
2.
4
2 ∙ 180°
∙
°
3.
5
3 ∙ 180°
∙
°
°
4.
6
∙
°
°
5.
n
∙
°
∙
°
∙
90°
°
Exercise Sets:
Homework:
p. 403: 1acegi, 2, 3ace, 4, 10, 11acegik, 12ace, 14ace, 19ac
Geogebra:
p. 403: 5ace, 6, 7ac, 8ace, 9aceg, 16, 17
Blazeview:
p. 403: 8, 13ac, 15, 36